GATE DA 2025 | Question 51

MSQ · 2 marks

Let \(f : \mathbb{R} \to \mathbb{R}\) be a twice-differentiable function and suppose its second derivative satisfies \(f''(x) > 0\) for all \(x \in \mathbb{R}\). Which of the following statements is/are ALWAYS correct?

  1. Option A: \(f\) has a local minima
  2. Option B: There does not exist \(x\) and \(y\), \(x \neq y\), such that \(f'(x) = f'(y) = 0\) (correct)
  3. Option C: \(f\) has at most one global minimum (correct)
  4. Option D: \(f\) has at most one local minimum (correct)

Answer

(B), (C), (D)

Explanation

Since \(f''(x) > 0\) everywhere, \(f'\) is strictly increasing, so it can be zero at most once; \(f\) is strictly convex.

  • A fails: \(f(x) = e^{x}\) has \(f''(x) = e^{x} > 0\), yet \(f'(x) = e^{x}\) is never 0, so this \(f\) has no minimum at all.
  • B holds: a strictly increasing \(f'\) cannot be 0 at two different points.
  • D holds: at a local minimum of a differentiable function \(f' = 0\), and there is at most one such point.
  • C holds: every global minimum is also a local minimum, so there is at most one.

Answer: B, C, D